Question

Consider the following reaction at \(1600^{\circ} \mathrm{C}\) : $$ \mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{Br}(g) $$ When 1.05 moles of \(\mathrm{Br}_{2}\) are put in a 0.980 - \(\mathrm{L}\) flask, 1.20 percent of the \(\mathrm{Br}_{2}\) undergoes dissociation. Calculate the equilibrium constant \(K_{\mathrm{c}}\) for the reaction.

Short answer

The equilibrium constant \(K_c\) for the reaction is 0.00062.

Step-by-step solution

Calculate initial concentrations

Start by calculating the initial concentration of \(\mathrm{Br}_{2}\). The initial concentration of \(\mathrm{Br}_{2}\) can be calculated by dividing the original number of moles (1.05 moles) by the volume of the flask (0.980 L): \[ [\mathrm{Br}_{2}]_{\text{initial}} = \frac{1.05 \, \text{moles}}{0.980 \, \text{L}} = 1.07 \, M \] At the start, there is no Br (before any dissociation happens), so its initial concentration is 0 M.

Calculate the concentration of dissociated \(\mathrm{Br}_{2}\)

Since 1.20% of the \(\mathrm{Br}_{2}\) undergoes dissociation, the concentration of dissociated \(\mathrm{Br}_{2}\) can be calculated by multiplying \([Br_{2}]\)initial by 0.0120 (which is 1.20% expressed as a decimal): \[\Delta [\mathrm{Br}_{2}] = [\mathrm{Br}_{2}]_{\text{initial}} \times 0.0120 = 1.07 \, M \times 0.0120 = 0.0128 \, M\] This means that at equilibrium, the concentration of \(\mathrm{Br}_{2}\) (\([Br_{2}]\)eq) is given by:\[[\mathrm{Br}_{2}]_{\text{eq}} = [\mathrm{Br}_{2}]_{\text{initial}} - \Delta [\mathrm{Br}_{2}] = 1.07 \, M - 0.0128 \, M = 1.0572 \, M\]

Calculate the concentration of \(\mathrm{Br}\)

For every one mole of \(\mathrm{Br}_{2}\) that dissociates, two moles of Br are created. Therefore, the change in the concentration of \(\mathrm{Br}\) is twice the change in the concentration of \(\mathrm{Br}_{2}\):\[\Delta [Br] = 2 \times \Delta [\mathrm{Br}_{2}] = 2 \times 0.0128 \, M = 0.0256 \, M\] Therefore, the equilibrium concentration of \(\mathrm{Br}\) is equal to the change in its concentration (since it started at 0 M):\[[Br]_{\text{eq}} = \Delta [Br] = 0.0256 \, M\]

Calculate equilibrium constant

Now use the equilibrium concentrations calculated above to compute the equilibrium constant \(K_c\). The expression for \(K_c\) for the reaction is \[ K_c = \frac{[Br]^2}{[\mathrm{Br}_{2}]});\] Substitute the equilibrium concentrations into this expression to obtain:\[ K_c = \frac{(0.0256 \, M)^2}{1.0572 \, M} = 0.00062\]

Key concepts

Chemical Equilibrium

Chemical equilibrium refers to a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. The quantities of reactants and products remain constant over time, not necessarily equal. It's essential to understand that at equilibrium, the chemical processes continue to occur; however, the overall concentrations of reactants and products do not change. This is a dynamic condition, and it’s crucial when predicting the outcome of a reaction. It helps in understanding how to manipulate conditions, such as temperature and concentrations, to favor the production of desired substances.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of a reactant or product in a chemical reaction at equilibrium. When we say 'concentration,' we're talking about how much of a substance is in a certain volume of a solution. In the classroom, students learn to calculate this by applying the ICE method (Initial, Change, Equilibrium) to set up an equilibrium table for a reaction. This method illustrates how the reactants' concentrations change as they form products, which in turn may recombine to form reactants again. Quantifying the equilibrium concentration is crucial when determining the equilibrium constant, which predicts the extent of a reaction under a set of given conditions.

Chemical Reaction

A chemical reaction is a process that transforms one set of chemical substances into another. Classically, these reactions involve changes in the structure of molecules and can result in the formation of new products with different properties from the original substances. Reactions may release or absorb energy, and they occur everywhere: in living cells, laboratories, and industrial processes. Balancing chemical equations is a foundational skill in chemistry, ensuring that the same number of atoms of each element is present on both sides of the equation to comply with the law of conservation of mass.